The ability to generate explanations that are understood by explainees is the quintessence of explainable artificial intelligence. Since understanding depends on the explainee's background and needs, recent research focused on co-constructive explanation dialogues, where an explainer continuously monitors the explainee's understanding and adapts their explanations dynamically. We investigate the ability of large language models (LLMs) to engage as explainers in co-constructive explanation dialogues. In particular, we present a user study in which explainees interact with an LLM in two settings, one of which involves the LLM being instructed to explain a topic co-constructively. We evaluate the explainees' understanding before and after the dialogue, as well as their perception of the LLMs' co-constructive behavior. Our results suggest that LLMs show some co-constructive behaviors, such as asking verification questions, that foster the explainees' engagement and can improve understanding of a topic. However, their ability to effectively monitor the current understanding and scaffold the explanations accordingly remains limited.